FYSA2020 Mechanics (5 cr)

Study level:
Intermediate studies
Grading scale:
Responsible organisation:
Department of Physics
Curriculum periods:
2020-2021, 2021-2022, 2022-2023, 2023-2024


  • Newtonian mechanics (Newton’s laws and conservation laws)

  • Oscillations (linear, damped, two-dimensional)

  • calculus of variations (Euler-Lagrange equations)

  • Lagrangian mechanics (Lagrangian, generalized coordinates, constraints, connection between conservation laws and symmetries, basics of Hamiltonian mechanics)

  • gravitation and central-field motion (Newton’s law of gravitation, central-field motion, effective potential, planetary motion)

  • Newtonian mechanics and selected types of problems

  • non-intertial coordinate systems (rotating coordinate systems, centrifugal force and Coriolis force)

  • rigid-body dynamics (energy and angular momentum, moments of inertia, principal axes, equations of motion for rotations, stability of rotations)

  • coupled oscillations (general procedure of solving, applications)

  • methods for solving various types of problems (lecture examples and problem sets) 

Learning outcomes

After completing the study unit the student can

  • describe the foundations of Newtonian mechanics

  • apply Newtonian mechanics in three-dimensional situations

  • decide when Lagrangian or Hamiltonian mechanics is particularly useful

  • apply Lagrangian mechanics in cases with constraints

  • reduce for central forces a two-body problem into one-body problem

  • describe planetary motion and the concept of constant of motion

  • deduce the apparent forces in a rotating corrdinate system

  • for the moment of inertia tensor for a rigid body

  • apply the equations of motion of a rigid body

  • find normal modes and their frequencies for coupled oscillations

  • solve applied problems staring from the equations of the basic theory 

Description of prerequisites

Basic physics studies:
  • FYSP1010 Introduction to Mechanics

  • FYSP1020 Waves and thermodynamics

Selected topics from courses in mathematics:
  • MATP211-213 Calculus 1-3, especially: 

    • Properties of elementary functions

    • Derivative and integral 

  • MATA114 Differential equation; especially: 

    • 1st order separable differential equation

    • 2st order constant-coefficient differential equation 

  • MATA181-181 Vector Calculus 1-2, especially: 

    • Differential, partial derivative, gradient

    • Integral over a three-dimensional volume

    • Integral over a path 

  • MATP121-122 Linear Algebraand Geometry 1-2; especially

    • Vectors: sum, dot product, cross product

    • Matrix and determinant

    • Products of vectors and matrices

    • Rotation matrices

    • Eigenvalues and eigenvectors 

  • In addition the basic properties of complex numbers. 

Study materials

Luentomateriaali TIM:ssä


  • Thornton & Marion: Classical dynamics of particles and systems. ISBN: 978-0-495-55610-7

Completion methods

Method 1

Ajankohta kevätlukukausi, 4. periodi
Evaluation criteria:
Harjoitukset 24/72, tentti 36/72 ja laboratoriotyöt 12/72.
Time of teaching:
Period 4
Select all marked parts

Method 2

This completion method is intended for students for whom method 1 is not possible for specific reasons (e.g. language, living elsewhere). The course lecturer must be contacted before enrolling on the course.
Evaluation criteria:
Exam 60/72, laboratory work 12/72
Select all marked parts
Parts of the completion methods

Teaching (5 cr)

Participation in teaching
Grading scale:
Evaluation criteria:
Harjoitukset 24/72, tentti 36/72 ja laboratoriotyöt 12/72.
Study methods:

Luennot, laskuharjoitukset, tentti ja laboratoriotyöt. 



Independent study (5 cr)

Independent study
Grading scale:
Evaluation criteria:
Exam 60/72, laboratory work 12/72.
Study methods:

Self-study, exam and laboratory work.